I believe that you can echo the '.dump' command to SQLite.
echo .dump | sqlite3 db | ssh user@rsync.net "dd of=db_dump"
It worked for me. echo .dump | sqlite3 db | ssh user@rsync.net "dd of=db_dump"
It worked for me.sqlite3 db .dump | ...
I have added that one-liner to the remote commands page:
https://www.rsync.net/resources/howto/remote_commands.html
... although based on some of your siblings, I changed .dump to .backup ...
As noted '.dump' will work in your pipeline, but with the caveats I and others noted elsewhere.